LMFDB - Newform orbit 627.2.i.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [627,2,Mod(463,627)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(627, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("627.463");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$627 = 3 \cdot 11 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	627.i (of order $3$ , degree $2$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.00662020673$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{14} - x^{13} + 12 x^{12} - 5 x^{11} + 94 x^{10} - 34 x^{9} + 375 x^{8} - 46 x^{7} + 1040 x^{6} + \cdots + 9$ x^14 - x^13 + 12x^12 - 5x^11 + 94x^10 - 34x^9 + 375x^8 - 46x^7 + 1040x^6 - 63x^5 + 1295x^4 + 674x^3 + 793x^2 + 87x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{13}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{10} - 1) q^{3} + ( - \beta_{11} + \beta_{10} - \beta_{3}) q^{4} + ( - \beta_{11} - \beta_{9} + \cdots + \beta_1) q^{5} - \beta_1 q^{6} + ( - \beta_{7} - \beta_{2} + 1) q^{7}+ \cdots - \beta_{10} q^{99}+O(q^{100})$ q + (b2 + b1) * q^2 + (-b10 - 1) * q^3 + (-b11 + b10 - b3) * q^4 + (-b11 - b9 + b2 + b1) * q^5 - b1 * q^6 + (-b7 - b2 + 1) * q^7 + (-b8 - b7 - b6 - b2) * q^8 + b10 * q^9 + (b10 + b9 + b6 - b5) * q^10 - q^11 + (b3 + 1) * q^12 + (b13 + b12 - b11 + 2b10 - b9 - b8 - b7 - b6 + b5 - b3) q^13 + (-b12 + 2b11 - 2b10 - b5 + b4 - 2) * q^14 + (b11 + b9 + b6 + b3 - b1) * q^15 + (2b13 + b12 + b10 - 2b9 + b2 + b1 + 1) * q^16 + (b12 - b10 - b9 - 1) * q^17 - b2 * q^18 + (b8 + b6 + b5 - b4 + b1) * q^19 + (-b6 - 2b2 - 3) q^20 + (b12 - b10 + b2 + b1 - 1) * q^21 + (-b2 - b1) * q^22 + (b13 + 2b10 - 2b9 - b8 - 2b6 + b1) q^23 + (b13 + b12 - b9 + b2 + b1) * q^24 + (b13 + b11 + 2b10 - b8 - b5 + b3 - b1) q^25 + (-b7 + 2b6 + b4 - 2b2 + 3) * q^26 + q^27 + (-3b13 - 2b12 + b9 + 3b8 + 2b7 + b6 - 2b5 - 4b1) * q^28 + (-b13 - b11 + 3b10 - 2b9 + b8 - 2b6 - b3 + 2b1) * q^29 + (-b6 + b4 + 1) * q^30 + (b4 - 2) * q^31 + (-2b13 - b12 - 2b11 + 2b9 + 2b8 + b7 + 2b6 - b5 - 2b3) * q^32 + (b10 + 1) * q^33 + (-b13 - 3b10 + 2b9 + b8 + 2b6 - 2b1) * q^34 + (-b13 - 2b9 - 2b5 + 2b4) q^35 + (b11 - b10 - 1) * q^36 + (2b8 + b7 + b6 - b4 + 4) q^37 + (-b13 - b12 + b11 - 2b10 + 3b9 - b8 - b7 + b6 + b4 - 2b2 - 3b1 - 4) * q^38 + (b8 + b7 + b6 - b4 + b3 + 2) * q^39 + (b13 + b12 + b11 - 2b10 - b5 + b4 - b2 - b1 - 2) q^40 + (b12 - b10 + b5 - b4 + 2b2 + 2b1 - 1) * q^41 + (b12 - 2b11 + 2b10 - b7 + b5 - 2b3) q^42 + (-3b11 - b9 + b5 - b4) q^43 + (b11 - b10 + b3) * q^44 + (-b6 - b3 - b2) * q^45 + (3b8 + 3b7 + 4b6 - 2b4 + 2b2) q^46 + (b13 + b10 + 2b9 - b8 + 2b6 - b5) * q^47 + (-2b13 - b12 - b10 + 2b9 + 2b8 + b7 + 2b6 - b1) * q^48 + (-b8 - b7 - b6 - b4 + b3 + 2b2 + 4) q^49 + (3b8 + 3b7 - b4 - b3 - b2 + 1) * q^50 + (-b12 + b10 + b9 + b7 + b6) * q^51 + (-b13 - 2b12 + 2b11 - 4b10 + b9 - 2b5 + 2b4 - b2 - b1 - 4) q^52 + (-b13 + 2b11 - b10 + b8 - b5 + 2b3 - 3b1) q^53 + (b2 + b1) * q^54 + (b11 + b9 - b2 - b1) * q^55 + (-2b8 - 2b7 - 4b6 - b4 + 2b3 - 2b2 + 5) q^56 + (-b13 + b9 - b5 + b2) * q^57 + (3b6 - 2b4 + b3 - b2 - 1) * q^58 + (2b11 - 3b10 + 2b9 - 2b5 + 2b4 - b2 - b1 - 3) q^59 + (3b10 - b9 + 2b2 + 2b1 + 3) q^60 + (b13 + b11 - 5b10 + b9 - b8 + b6 + b3 - 2b1) * q^61 + (-b13 - b12 - b11 + b10 - b9 - b5 + b4 - b2 - b1 + 1) * q^62 + (-b12 + b10 + b7 - b1) * q^63 + (-b8 - 2b7 - 3b6 - 6b2 - 2) q^64 + (-b8 - 2b7 - 3b6 + 3b4 - 2b3 - b2 - 2) * q^65 + b1 * q^66 + (2b13 - b12 + b11 + 2b9 - 2b8 + b7 + 2b6 + b5 + b3 + b1) * q^67 + (-3b8 - b7 - 2b6 + 2b4 + b3 - b2 + 1) q^68 + (b8 + 2b6 + b2 + 2) q^69 + (-3b13 - 3b12 + b11 - 3b10 + 2b9 + 3b8 + 3b7 + 2b6 - 4b5 + b3 - 2b1) q^70 + (b13 - b12 - b11 - 2b10 + 2b9 + 3b5 - 3b4 - 2) * q^71 + (-b13 - b12 + b9 + b8 + b7 + b6 - b1) * q^72 + (4b13 + 3b12 - b11 + 2b5 - 2b4 + b2 + b1) * q^73 + (-2b13 - b12 - b11 - 2b10 + 3b9 + b5 - b4 + 2b2 + 2b1 - 2) q^74 + (b8 + b4 - b3 - b2 + 2) * q^75 + (3b11 - b10 - 2b9 - b8 - 2b7 - 3b6 - b5 + b4 + 2b3 - 2b2 - b1 + 2) * q^76 + (b7 + b2 - 1) * q^77 + (b12 - 3b10 + 2b9 + b5 - b4 + 2b2 + 2b1 - 3) * q^78 + (-2b12 + b11 + b10 - 3b9 + b5 - b4 + 2b2 + 2b1 + 1) * q^79 + (-3b13 - 2b12 - 2b11 + 4b10 - 2b9 + 3b8 + 2b7 - 2b6 - 2b3 + 2b1) * q^80 + (-b10 - 1) * q^81 + (b13 + 2b12 - 2b11 + 4b10 + b9 - b8 - 2b7 + b6 + 2b5 - 2b3 - b1) * q^82 + (3b8 + 2b7 - 2b4 + b3 + b2 + 3) q^83 + (-3b8 - 2b7 - b6 + 2b4 - 4b2) * q^84 + (2b13 + 2b10 - 2b8 + 3b5 - 2b1) q^85 + (3b13 + 3b12 + 2b11 - 3b10 - 3b8 - 3b7 + 2b3 + 3b1) * q^86 + (-b8 + 2b6 + b3 + 2b2 + 3) * q^87 + (b8 + b7 + b6 + b2) * q^88 + (b13 + b12 - 4b11 + 2b10 - b9 - b8 - b7 - b6 + 2b5 - 4b3) * q^89 + (-b10 - b9 + b5 - b4 - 1) * q^90 + (b11 - b10 - 2b9 - 2b6 + b5 + b3 - 5b1) q^91 + (-3b13 - 2b12 - 2b11 + 6b9 + b5 - b4 - 5b2 - 5b1) * q^92 + (2b10 + b5 - b4 + 2) q^93 + (-5b6 + b4 - 4b3 - 6b2 - 6) q^94 + (2b12 - b9 - 2b8 - 3b7 + b6 - 3b1 + 1) * q^95 + (-2b8 - b7 - 2b6 + b4 + 2b3) q^96 + (b13 - 2b11 + 6b10 - b9 + b5 - b4 + 4b2 + 4b1 + 6) * q^97 + (4b13 + 3b12 + 7b10 - 2b9 + b5 - b4 + 6b2 + 6b1 + 7) * q^98 - b10 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$14 q - q^{2} - 7 q^{3} - 9 q^{4} - 2 q^{5} - q^{6} + 14 q^{7} + 12 q^{8} - 7 q^{9} - 13 q^{10} - 14 q^{11} + 18 q^{12} - 8 q^{13} - 16 q^{14} - 2 q^{15} - 5 q^{16} - 9 q^{17} + 2 q^{18} - 14 q^{19} - 32 q^{20}+ \cdots + 7 q^{99}+O(q^{100})$ 14 * q - q^2 - 7 * q^3 - 9 * q^4 - 2 * q^5 - q^6 + 14 * q^7 + 12 * q^8 - 7 * q^9 - 13 * q^10 - 14 * q^11 + 18 * q^12 - 8 * q^13 - 16 * q^14 - 2 * q^15 - 5 * q^16 - 9 * q^17 + 2 * q^18 - 14 * q^19 - 32 * q^20 - 7 * q^21 + q^22 - 4 * q^23 - 6 * q^24 - 13 * q^25 + 38 * q^26 + 14 * q^27 - 20 * q^28 - 18 * q^29 + 26 * q^30 - 22 * q^31 - 18 * q^32 + 7 * q^33 + 10 * q^34 + 3 * q^35 - 9 * q^36 + 34 * q^37 - 28 * q^38 + 16 * q^39 - 14 * q^40 - 11 * q^41 - 16 * q^42 + 9 * q^44 + 4 * q^45 - 52 * q^46 - 13 * q^47 - 5 * q^48 + 60 * q^49 - 6 * q^50 - 9 * q^51 - 21 * q^52 + 2 * q^53 - q^54 + 2 * q^55 + 108 * q^56 + q^57 - 38 * q^58 - 12 * q^59 + 16 * q^60 + 35 * q^61 + 12 * q^62 - 7 * q^63 + 4 * q^64 + 4 * q^65 + q^66 + 7 * q^67 + 60 * q^68 + 8 * q^69 - 3 * q^70 - 19 * q^71 - 6 * q^72 - 14 * q^73 - 3 * q^74 + 26 * q^75 + 57 * q^76 - 14 * q^77 - 19 * q^78 - 11 * q^79 - 31 * q^80 - 7 * q^81 - 29 * q^82 + 18 * q^83 + 40 * q^84 - q^85 + 34 * q^86 + 36 * q^87 - 12 * q^88 - 11 * q^89 - 13 * q^90 + 13 * q^91 + 31 * q^92 + 11 * q^93 - 52 * q^94 + 10 * q^95 + 36 * q^96 + 33 * q^97 + 25 * q^98 + 7 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{14} - x^{13} + 12 x^{12} - 5 x^{11} + 94 x^{10} - 34 x^{9} + 375 x^{8} - 46 x^{7} + 1040 x^{6} + \cdots + 9$ x^14 - x^13 + 12*x^12 - 5*x^11 + 94*x^10 - 34*x^9 + 375*x^8 - 46*x^7 + 1040*x^6 - 63*x^5 + 1295*x^4 + 674*x^3 + 793*x^2 + 87*x + 9 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 12492522413 \nu^{13} - 204368968522 \nu^{12} + 542757010127 \nu^{11} + \cdots + 23712717353469 ) / 229458856979425$ (12492522413v^13 - 204368968522v^12 + 542757010127v^11 - 2353036102176v^10 + 4173865794440v^9 - 17208242514637v^8 + 28327934537566v^7 - 63367221095436v^6 + 82251019929668v^5 - 157393350709793v^4 + 248597933798784v^3 - 130927359258800v^2 - 14432052033516*v + 23712717353469) / 229458856979425
$\beta_{3}$	$=$	$( 191876446109 \nu^{13} - 392846741171 \nu^{12} + 2290573490111 \nu^{11} + \cdots - 688264138236558 ) / 229458856979425$ (191876446109v^13 - 392846741171v^12 + 2290573490111v^11 - 2999568687618v^10 + 16783496752595v^9 - 23643238632691v^8 + 62792565064438v^7 - 69258796620148v^6 + 156606321797774v^5 - 232420117273949v^4 + 139347319365162v^3 - 205120234672400v^2 - 22625867903538*v - 688264138236558) / 229458856979425
$\beta_{4}$	$=$	$( 233645909396 \nu^{13} + 1159637467626 \nu^{12} - 858485307716 \nu^{11} + \cdots - 78464232577577 ) / 229458856979425$ (233645909396v^13 + 1159637467626v^12 - 858485307716v^11 + 16343124230858v^10 - 10812659744195v^9 + 116412585089946v^8 - 148165858874528v^7 + 435778975801013v^6 - 477029759614594v^5 + 1035309595717144v^4 - 1164335685425297v^3 + 839764902394400v^2 + 92538468052878*v - 78464232577577) / 229458856979425
$\beta_{5}$	$=$	$( - 1254063459263 \nu^{13} + 2622520781297 \nu^{12} - 21134211907002 \nu^{11} + \cdots - 5542565905719 ) / 688376570938275$ (-1254063459263v^13 + 2622520781297v^12 - 21134211907002v^11 + 23164466874301v^10 - 160865974696040v^9 + 134454336375437v^8 - 748473556821516v^7 + 349246862361461v^6 - 1668839369484568v^5 + 554057795718918v^4 - 1868201158554334v^3 - 648581909765050v^2 + 2353484150507191*v - 5542565905719) / 688376570938275
$\beta_{6}$	$=$	$( - 1437608426327 \nu^{13} + 2093175839863 \nu^{12} - 16568600828333 \nu^{11} + \cdots + 400858213276249 ) / 458917713958850$ (-1437608426327v^13 + 2093175839863v^12 - 16568600828333v^11 + 19222033122129v^10 - 124765184864285v^9 + 153529930329298v^8 - 433934353303414v^7 + 612166862161169v^6 - 1064761370508497v^5 + 1526981437639097v^4 - 853875651955236v^3 + 1359647717163200v^2 + 149991457096839*v + 400858213276249) / 458917713958850
$\beta_{7}$	$=$	$( - 1050777220112 \nu^{13} + 3824944080053 \nu^{12} - 16370513374848 \nu^{11} + \cdots + 931396499850044 ) / 229458856979425$ (-1050777220112v^13 + 3824944080053v^12 - 16370513374848v^11 + 37402968774449v^10 - 122271680406985v^9 + 281802233878763v^8 - 579516985496284v^7 + 997180563884414v^6 - 1558103303218757v^5 + 2655790430296982v^4 - 2575217695624991v^3 + 2266279015489200v^2 + 249886148067459*v + 931396499850044) / 229458856979425
$\beta_{8}$	$=$	$( 3414237642421 \nu^{13} - 7699374314749 \nu^{12} + 43882057476759 \nu^{11} + \cdots - 25\!\cdots\!27 ) / 458917713958850$ (3414237642421v^13 - 7699374314749v^12 + 43882057476759v^11 - 70497609649267v^10 + 327569887733855v^9 - 545051972940454v^8 + 1309688978920322v^7 - 1972855778975637v^6 + 3358457777649331v^5 - 5264628791135131v^4 + 3977249419176228v^3 - 4582932155553600v^2 - 505443232896597*v - 2500778386511027) / 458917713958850
$\beta_{9}$	$=$	$( - 14394158081609 \nu^{13} + 19182465543746 \nu^{12} - 173373160679511 \nu^{11} + \cdots - 12\!\cdots\!92 ) / 13\!\cdots\!50$ (-14394158081609v^13 + 19182465543746v^12 - 173373160679511v^11 + 112551866288368v^10 - 1337015451365945v^9 + 773249574686741v^8 - 5327812494518913v^7 + 1208086609706648v^6 - 14916808336547749v^5 + 1779817940926674v^4 - 18448081063430662v^3 - 11864123953379950v^2 - 11580892154463137*v - 1271057045255292) / 1376753141876550
$\beta_{10}$	$=$	$( 7904239117823 \nu^{13} - 7941716685062 \nu^{12} + 95463976319442 \nu^{11} + \cdots + 42588388412874 ) / 688376570938275$ (7904239117823v^13 - 7941716685062v^12 + 95463976319442v^11 - 41149466619496v^10 + 750057585381890v^9 - 281265727389302v^8 + 3015714396727536v^7 - 448578803032556v^6 + 8410510345822228v^5 - 744720124211853v^4 + 10708169709710164v^3 + 4581663364016350v^2 + 6660843698210039*v + 42588388412874) / 688376570938275
$\beta_{11}$	$=$	$( 7904239117823 \nu^{13} - 7941716685062 \nu^{12} + 95463976319442 \nu^{11} + \cdots + 730964959351149 ) / 229458856979425$ (7904239117823v^13 - 7941716685062v^12 + 95463976319442v^11 - 41149466619496v^10 + 750057585381890v^9 - 281265727389302v^8 + 3015714396727536v^7 - 448578803032556v^6 + 8410510345822228v^5 - 744720124211853v^4 + 10708169709710164v^3 + 4811122220995775v^2 + 6660843698210039*v + 730964959351149) / 229458856979425
$\beta_{12}$	$=$	$( 38616818188369 \nu^{13} - 35376655705411 \nu^{12} + 457197545552226 \nu^{11} + \cdots + 29\!\cdots\!22 ) / 688376570938275$ (38616818188369v^13 - 35376655705411v^12 + 457197545552226v^11 - 155047998241163v^10 + 3582859190445295v^9 - 1041466844057281v^8 + 14107692317713908v^7 - 755681755605718v^6 + 38875164204800309v^5 + 93789059762916v^4 + 46316777454149567v^3 + 29096545730067950v^2 + 27351737265885817*v + 2998793902155522) / 688376570938275
$\beta_{13}$	$=$	$( - 93208391901109 \nu^{13} + 95995182831046 \nu^{12} + \cdots - 79\!\cdots\!92 ) / 13\!\cdots\!50$ (-93208391901109v^13 + 95995182831046v^12 - 1116292080830211v^11 + 485721610746068v^10 - 8730666442879645v^9 + 3317463924881791v^8 - 34755630046668513v^7 + 4362795660847648v^6 - 96456658751052899v^5 + 5659244272618674v^4 - 119169490599915962v^3 - 67178097973759700v^2 - 72834655743270637*v - 7990387698260292) / 1376753141876550

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{11} - 3\beta_{10} - 3$ b11 - 3*b10 - 3
$\nu^{3}$	$=$	$\beta_{8} + \beta_{7} + \beta_{6} + 5\beta_{2}$ b8 + b7 + b6 + 5*b2
$\nu^{4}$	$=$	$- 2 \beta_{13} - \beta_{12} - 6 \beta_{11} + 13 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \beta_{7} + \cdots - \beta_1$ -2b13 - b12 - 6b11 + 13b10 + 2b9 + 2b8 + b7 + 2b6 - 6*b3 - b1
$\nu^{5}$	$=$	$-10\beta_{13} - 9\beta_{12} - 2\beta_{11} + 10\beta_{9} - \beta_{5} + \beta_{4} - 28\beta_{2} - 28\beta_1$ -10b13 - 9b12 - 2b11 + 10b9 - b5 + b4 - 28b2 - 28b1
$\nu^{6}$	$=$	$-21\beta_{8} - 12\beta_{7} - 23\beta_{6} + 36\beta_{3} - 16\beta_{2} + 64$ -21b8 - 12b7 - 23b6 + 36b3 - 16*b2 + 64
$\nu^{7}$	$=$	$80 \beta_{13} + 68 \beta_{12} + 26 \beta_{11} - 11 \beta_{10} - 82 \beta_{9} - 80 \beta_{8} + \cdots + 170 \beta_1$ 80b13 + 68b12 + 26b11 - 11b10 - 82b9 - 80b8 - 68b7 - 82b6 + 11b5 + 26b3 + 170*b1
$\nu^{8}$	$=$	$177 \beta_{13} + 109 \beta_{12} + 225 \beta_{11} - 347 \beta_{10} - 201 \beta_{9} + 3 \beta_{5} + \cdots - 347$ 177b13 + 109b12 + 225b11 - 347b10 - 201b9 + 3b5 - 3b4 + 170b2 + 170*b1 - 347
$\nu^{9}$	$=$	$600\beta_{8} + 491\beta_{7} + 630\beta_{6} - 89\beta_{4} - 252\beta_{3} + 1093\beta_{2} - 173$ 600b8 + 491b7 + 630b6 - 89b4 - 252b3 + 1093b2 - 173
$\nu^{10}$	$=$	$- 1393 \beta_{13} - 902 \beta_{12} - 1465 \beta_{11} + 2039 \beta_{10} + 1601 \beta_{9} + \cdots - 1535 \beta_1$ -1393b13 - 902b12 - 1465b11 + 2039b10 + 1601b9 + 1393b8 + 902b7 + 1601b6 - 50b5 - 1465b3 - 1535*b1
$\nu^{11}$	$=$	$- 4409 \beta_{13} - 3507 \beta_{12} - 2179 \beta_{11} + 1844 \beta_{10} + 4717 \beta_{9} - 649 \beta_{5} + \cdots + 1844$ -4409b13 - 3507b12 - 2179b11 + 1844b10 + 4717b9 - 649b5 + 649b4 - 7319b2 - 7319*b1 + 1844
$\nu^{12}$	$=$	$-10656\beta_{8} - 7149\beta_{7} - 12262\beta_{6} + 561\beta_{4} + 9869\beta_{3} - 12778\beta_{2} + 12776$ -10656b8 - 7149b7 - 12262b6 + 561b4 + 9869b3 - 12778b2 + 12776
$\nu^{13}$	$=$	$32226 \beta_{13} + 25077 \beta_{12} + 17760 \beta_{11} - 16756 \beta_{10} - 34954 \beta_{9} + \cdots + 50450 \beta_1$ 32226b13 + 25077b12 + 17760b11 - 16756b10 - 34954b9 - 32226b8 - 25077b7 - 34954b6 + 4552b5 + 17760b3 + 50450*b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/627\mathbb{Z}\right)^\times$ .

$n$	$343$	$419$	$496$
$\chi(n)$	$1$	$1$	$\beta_{10}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

463.1

1.35831	−	2.35267i
0.914361	−	1.58372i
0.794699	−	1.37646i
−0.0560088	+	0.0970101i
−0.371378	+	0.643245i
−1.01277	+	1.75417i
−1.12721	+	1.95239i
1.35831	+	2.35267i
0.914361	+	1.58372i
0.794699	+	1.37646i
−0.0560088	−	0.0970101i
−0.371378	−	0.643245i
−1.01277	−	1.75417i
−1.12721	−	1.95239i

−1.35831

−

2.35267i

−0.500000

−

0.866025i

−2.69004

4.65928i

−0.467652

−

0.809997i

−1.35831

2.35267i

4.17977

9.18241

−0.500000

0.866025i

−1.27044

2.20046i

463.2

−0.914361

−

1.58372i

−0.500000

−

0.866025i

−0.672111

1.16413i

1.20135

2.08079i

−0.914361

1.58372i

2.48304

−1.19924

−0.500000

0.866025i

2.19693

−

3.80519i

463.3

−0.794699

−

1.37646i

−0.500000

−

0.866025i

−0.263092

0.455690i

−1.98856

−

3.44428i

−0.794699

1.37646i

0.0905430

−2.34248

−0.500000

0.866025i

−3.16061

5.47434i

463.4

0.0560088

0.0970101i

−0.500000

−

0.866025i

0.993726

−

1.72118i

−1.02085

−

1.76816i

0.0560088

−

0.0970101i

−3.03154

0.446665

−0.500000

0.866025i

0.114353

−

0.198065i

463.5

0.371378

0.643245i

−0.500000

−

0.866025i

0.724157

−

1.25428i

−1.26435

−

2.18992i

0.371378

−

0.643245i

4.12826

2.56125

−0.500000

0.866025i

0.939105

−

1.62658i

463.6

1.01277

1.75417i

−0.500000

−

0.866025i

−1.05142

1.82111i

1.77847

3.08040i

1.01277

−

1.75417i

3.36717

−0.208293

−0.500000

0.866025i

−3.60237

6.23949i

463.7

1.12721

1.95239i

−0.500000

−

0.866025i

−1.54123

2.66948i

0.761596

1.31912i

1.12721

−

1.95239i

−4.21725

−2.44032

−0.500000

0.866025i

−1.71696

2.97387i

562.1

−1.35831

2.35267i

−0.500000

0.866025i

−2.69004

−

4.65928i

−0.467652

0.809997i

−1.35831

−

2.35267i

4.17977

9.18241

−0.500000

−

0.866025i

−1.27044

−

2.20046i

562.2

−0.914361

1.58372i

−0.500000

0.866025i

−0.672111

−

1.16413i

1.20135

−

2.08079i

−0.914361

−

1.58372i

2.48304

−1.19924

−0.500000

−

0.866025i

2.19693

3.80519i

562.3

−0.794699

1.37646i

−0.500000

0.866025i

−0.263092

−

0.455690i

−1.98856

3.44428i

−0.794699

−

1.37646i

0.0905430

−2.34248

−0.500000

−

0.866025i

−3.16061

−

5.47434i

562.4

0.0560088

−

0.0970101i

−0.500000

0.866025i

0.993726

1.72118i

−1.02085

1.76816i

0.0560088

0.0970101i

−3.03154

0.446665

−0.500000

−

0.866025i

0.114353

0.198065i

562.5

0.371378

−

0.643245i

−0.500000

0.866025i

0.724157

1.25428i

−1.26435

2.18992i

0.371378

0.643245i

4.12826

2.56125

−0.500000

−

0.866025i

0.939105

1.62658i

562.6

1.01277

−

1.75417i

−0.500000

0.866025i

−1.05142

−

1.82111i

1.77847

−

3.08040i

1.01277

1.75417i

3.36717

−0.208293

−0.500000

−

0.866025i

−3.60237

−

6.23949i

562.7

1.12721

−

1.95239i

−0.500000

0.866025i

−1.54123

−

2.66948i

0.761596

−

1.31912i

1.12721

1.95239i

−4.21725

−2.44032

−0.500000

−

0.866025i

−1.71696

−

2.97387i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	627.2.i.e	✓	14
19.c	even	3	1	inner	627.2.i.e	✓	14

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
627.2.i.e	✓	14	1.a	even	1	1	trivial
627.2.i.e	✓	14	19.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(627, [\chi])$ :

T_{2}^{14} + T_{2}^{13} + 12 T_{2}^{12} + 5 T_{2}^{11} + 94 T_{2}^{10} + 34 T_{2}^{9} + 375 T_{2}^{8} + \cdots + 9

T_{5}^{14} + 2 T_{5}^{13} + 26 T_{5}^{12} + 32 T_{5}^{11} + 429 T_{5}^{10} + 507 T_{5}^{9} + \cdots + 62500

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{14} + T^{13} + \cdots + 9$ T^14 + T^13 + 12T^12 + 5T^11 + 94T^10 + 34T^9 + 375T^8 + 46T^7 + 1040T^6 + 63T^5 + 1295T^4 - 674T^3 + 793T^2 - 87T + 9
$3$	$(T^{2} + T + 1)^{7}$ (T^2 + T + 1)^7
$5$	$T^{14} + 2 T^{13} + \cdots + 62500$ T^14 + 2T^13 + 26T^12 + 32T^11 + 429T^10 + 507T^9 + 3510T^8 + 2926T^7 + 18885T^6 + 14857T^5 + 56428T^4 + 23965T^3 + 94119T^2 + 53250*T + 62500
$7$	$(T^{7} - 7 T^{6} + \cdots - 167)^{2}$ (T^7 - 7T^6 - 15T^5 + 188T^4 - 159T^3 - 1120T^2 + 1947T - 167)^2
$11$	$(T + 1)^{14}$ (T + 1)^14
$13$	$T^{14} + 8 T^{13} + \cdots + 400$ T^14 + 8T^13 + 84T^12 + 314T^11 + 2574T^10 + 9525T^9 + 48665T^8 + 82354T^7 + 164353T^6 + 84938T^5 + 213357T^4 + 115592T^3 + 148116T^2 - 7520*T + 400
$17$	$T^{14} + 9 T^{13} + \cdots + 242064$ T^14 + 9T^13 + 90T^12 + 375T^11 + 2335T^10 + 7092T^9 + 41104T^8 + 87903T^7 + 385855T^6 + 526836T^5 + 2444366T^4 + 2707716T^3 + 2502601T^2 + 879204*T + 242064
$19$	$T^{14} + \cdots + 893871739$ T^14 + 14T^13 + 79T^12 + 237T^11 + 207T^10 - 3900T^9 - 37178T^8 - 196687T^7 - 706382T^6 - 1407900T^5 + 1419813T^4 + 30886077T^3 + 195611821T^2 + 658642334*T + 893871739
$23$	$T^{14} + 4 T^{13} + \cdots + 24964$ T^14 + 4T^13 + 104T^12 + 492T^11 + 8799T^10 + 33863T^9 + 221870T^8 + 38740T^7 + 948579T^6 - 886389T^5 + 4778998T^4 - 4457235T^3 + 5367151T^2 - 375566*T + 24964
$29$	$T^{14} + 18 T^{13} + \cdots + 4080400$ T^14 + 18T^13 + 288T^12 + 2620T^11 + 25205T^10 + 179127T^9 + 1347560T^8 + 6813308T^7 + 30517891T^6 + 68377217T^5 + 128774196T^4 - 62813831T^3 + 163988989T^2 + 24698540*T + 4080400
$31$	$(T^{7} + 11 T^{6} + \cdots + 2272)^{2}$ (T^7 + 11T^6 + 16T^5 - 207T^4 - 881T^3 - 719T^2 + 1588T + 2272)^2
$37$	$(T^{7} - 17 T^{6} + \cdots + 25138)^{2}$ (T^7 - 17T^6 + 44T^5 + 541T^4 - 2281T^3 - 4513T^2 + 16743T + 25138)^2
$41$	$T^{14} + 11 T^{13} + \cdots + 25887744$ T^14 + 11T^13 + 156T^12 + 943T^11 + 9400T^10 + 49529T^9 + 359946T^8 + 1225772T^7 + 5867843T^6 + 12517311T^5 + 59580083T^4 + 90672386T^3 + 231311332T^2 - 71059008*T + 25887744
$43$	$T^{14} + \cdots + 36773431696$ T^14 + 162T^12 + 666T^11 + 19655T^10 + 72260T^9 + 1046865T^8 + 3547835T^7 + 38866681T^6 + 107966528T^5 + 704580853T^4 + 1323451598T^3 + 7831215737T^2 + 12602921844T + 36773431696
$47$	$T^{14} + \cdots + 521756964$ T^14 + 13T^13 + 278T^12 + 1945T^11 + 32998T^10 + 210288T^9 + 2472452T^8 + 9941649T^7 + 87890992T^6 + 312240104T^5 + 2110918691T^4 + 3823705271T^3 + 5917230811T^2 + 1907329842*T + 521756964
$53$	$T^{14} + \cdots + 566059264$ T^14 - 2T^13 + 159T^12 + 356T^11 + 19521T^10 + 24033T^9 + 631241T^8 + 756384T^7 + 14949435T^6 + 13931398T^5 + 152512230T^4 + 125734572T^3 + 1136916337T^2 + 779782800*T + 566059264
$59$	$T^{14} + \cdots + 1139737600$ T^14 + 12T^13 + 256T^12 + 1674T^11 + 30972T^10 + 209747T^9 + 1984557T^8 + 8694480T^7 + 57019535T^6 + 204498456T^5 + 1064357481T^4 + 2292920308T^3 + 6017751984T^2 - 2363605120*T + 1139737600
$61$	$T^{14} + \cdots + 20194683664$ T^14 - 35T^13 + 765T^12 - 11002T^11 + 120390T^10 - 995047T^9 + 6619150T^8 - 34770227T^7 + 156057152T^6 - 585741279T^5 + 2078669301T^4 - 5960698024T^3 + 14936437740T^2 - 20254369024*T + 20194683664
$67$	$T^{14} + \cdots + 143747139600$ T^14 - 7T^13 + 323T^12 - 1988T^11 + 75368T^10 - 393003T^9 + 6541674T^8 + 3164823T^7 + 187331948T^6 + 714234103T^5 + 6943676033T^4 + 24199811852T^3 + 79611194116T^2 + 117926189040*T + 143747139600
$71$	$T^{14} + \cdots + 22895650803600$ T^14 + 19T^13 + 586T^12 + 5925T^11 + 140953T^10 + 1217068T^9 + 21833550T^8 + 109295107T^7 + 1536121789T^6 + 5274369988T^5 + 76199247380T^4 + 128610165512T^3 + 1570209472729T^2 - 984802856220*T + 22895650803600
$73$	$T^{14} + \cdots + 67602080016$ T^14 + 14T^13 + 511T^12 + 5160T^11 + 147644T^10 + 1384562T^9 + 24925112T^8 + 173867123T^7 + 2527310368T^6 + 15625965285T^5 + 142906677809T^4 + 336145393332T^3 + 654365740996T^2 + 226316841744*T + 67602080016
$79$	$T^{14} + \cdots + 3037686466816$ T^14 + 11T^13 + 518T^12 + 2357T^11 + 143517T^10 + 456642T^9 + 24414372T^8 - 3033857T^7 + 2509678137T^6 - 1774289252T^5 + 168433615342T^4 - 370039548044T^3 + 5409655259265T^2 + 3890086356432*T + 3037686466816
$83$	$(T^{7} - 9 T^{6} + \cdots + 316262)^{2}$ (T^7 - 9T^6 - 196T^5 + 1199T^4 + 11249T^3 - 39809T^2 - 165657T + 316262)^2
$89$	$T^{14} + \cdots + 1942042919184$ T^14 + 11T^13 + 379T^12 + 980T^11 + 63356T^10 + 26581T^9 + 7517952T^8 - 19459739T^7 + 532371682T^6 - 1875876413T^5 + 26935160389T^4 - 122057481856T^3 + 578743682380T^2 - 1142026679712*T + 1942042919184
$97$	$T^{14} + \cdots + 5784058809$ T^14 - 33T^13 + 814T^12 - 10143T^11 + 102850T^10 - 494366T^9 + 2832001T^8 - 3460480T^7 + 70422398T^6 + 11614953T^5 + 738547843T^4 + 901304792T^3 + 6250376803T^2 + 5734928571*T + 5784058809
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 463.7
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	627.2.i.e

Level	$627$
Weight	$2$
Character orbit	627.i
Analytic conductor	$5.007$
Analytic rank	$0$
Dimension	$14$
Inner twists	$2$